A Dynamic Formation Procedure of Information Flow Networks

2.1 Equilibrium Networks in the Static Case

Let N = {1, 2, ⋯, n} with n ≥ 3 be the set of players and let i and j be typical members of this set. Each player chooses a level of personal information acquisition x_i ∈ X = [0, +∞) and a set of links with others to access their information, which is represented as a (row) vector:

where g_ii = 0, ∀i ∈ N and g_ij ∈ {0, 1}, for each j ∈ N\{i}. We say that player i has a link with player j if g_ij = 1, and the cost of linking with one other person is k > 0. Otherwise we have g_ij = 0. Our paper assumes that the cost of linking with one other person is homogeneous, and the link between player i and j allow both players share information. The set of strategies of player i is denoted by S_i = X × G_i. Define S = S₁ × S₂ × ⋯ × S_n as the set of strategies of all players. A strategy profile s = (x, g) ∈ S specifies the personal information acquired by each player x = (x₁, x₂, ⋯, x_n), and the network of relations (connection Matrix) g = (g₁, g₂, ⋯, g_n)^T, where T specifies a transposition.

The network g is a directed graph, where the arrow from i to j specifies g_ij = 1. Let G be the set of all possible directed graphs on n vertices. Define N_i(g) = {j ∈ N : g_ij = 1} as the set of players with whom i has formed a link. Let η_i(g) = | N_i(g) |. The closure of g is an undirected network denoted by g = cl(g), where g_ij = max{g_ij, g_ji}. In words, the closure of a directed network involves replacing every directed edge of by an undirected one. Define N_i(g) = {j ∈ N : g_ij = 1} as the set of players directly connected to i. The undirected link between two players reflects bilateral information exchange between them.

The payoff to player i under strategy profile s = (x, g) is

where c_i > 0 is the cost of information. We assume the cost of information that personally acquired is heterogeneous. We also assume that f(y) is twice continuously differentiable, increasing, and strictly concave in y. To focus on interesting cases we assume

and

Under these assumptions, there exists a number ŷ_i > 0 such that

i.e., ŷ_i solves f′(ŷ_i) = c_i.

A Nash equilibrium is a strategy profile s^* = (x^*, g^*) satisfying

Example 1

Let N = {1, 2, 3, 4, 5, 6} be the set of players, the cost of linking with one other person be homogeneous k = 15, the cost of information that personally acquired be heterogeneous and denoted by c₁ = c₂ = c₃ = c₄ = 12, c₅ = c₆ = 1021. Suppose that the payoffs are given by (1), where f(y) = ln(1 + y). We can check that s^* = (x^*, g^*) is a Nash equilibrium, where x∗=(x1∗,x2∗,x3∗,x4∗,x5∗,,x6∗)=(1270,1270,1270,1270,2970,2970), the corresponding connection matrix and the network structure are given below:

g∗=g1∗g2∗g3∗g4∗g5∗g6∗=000011000011000011000011000000000000

Figure 1

A network

Notice that equilibrium network in this example has typical “two-core-periphery” architecture. Player 5 and Player 6 become two hubs because they have slightly lower costs of acquiring information, the information that they acquire personally is 2970. Players 1, 2, 3 and 4 become spokes. Because the cost of linking with one other person is low, so the information that they acquire personally is 1270(x1∗=x2∗=x3∗=x4∗=1270) and aggregate information is 1 through linking with two hubs, in fact ŷ₁ = ŷ₂ = ŷ₃ = ŷ₄ = 1 at this moment, the aggregate information which Player 5 and Player 6 own is 1110 = ŷ₅ = ŷ₆. Under the equilibrium network, the payoff of spokes is

Πi(s∗)=ln⁡(1+y^i)−cixi∗−2k=ln⁡(1+1)−12×635−2×15=0.2074,i=1,2,3,4

However, when spokes acquire the optimal aggregate information 1 personally and do not link with hubs, the payoff is

Πi(s)=ln⁡(1+y^i)−ci=ln⁡(1+1)−12=0.193,i=1,2,3,4

Similarly, we can verify Π_i (s) < Π_i (s^*), ∀s ∈ S_i, ∀i ∈ N. So the spokes form links with two hubs while do not form links between them and two hubs do not form links between them is the equilibrium strategy for the players.

In this example, there is no link between two hubs because of k > c₅ŷ₆ = c₆ŷ₅ (0.2 > 1021 × 2970 = 0.1973). Spokes choose to form links with hubs because of k < c_iŷ₅ = c_iŷ₆ (i = 1, 2, 3, 4), i.e. 0.2 < 12 × 2970 = 0.2071.

As we see in Example 1, the cost heterogeneity of information acquisition is based on two levels. The primary cause lies in the high complexity of the algorithm of general equilibrium network. In fact, slight cost difference can help us distinguish which players attain information actively and which players become spokes.

To simplify symbolic system, we mark ŷ₁ and ŷ as the optimal value of information, which are obtained by players who have information cost advantage and have not advantage. We use c~ = c − ε < c to express the slight advantage of attaining information cost, where ε > 0 is a small number.

In the network g with core-periphery architecture, we assume that N_c(g) be the set of hubs, where | N_c(g)| = m, then N\N_c(g) be the set of spokes, and we have | N\N_c(g)| = mq, where q ∈ N₊, namely n = (q + 1)m. The homogeneous cost of linking with one other person is k which satisfies f(ŷ) − c ŷ < f(mŷ₁) − mk.

The proof of this lemma is similar to Lemma 1 in [1].

In fact, the first inequality in term 3) assures that spokes are not being connected and spokes are favorable to link with hubs; the second inequality assures that hubs are not being connected; the third inequality assures that spokes link with hubs if they acquire partial information actively. Meeting with information level of terms 1) and 2) can guarantee hubs owning the aggregate information ŷ₁ in the network, while the aggregate information owned by spokes is ŷ. It is shown in Example 1.

Similar to Theorem 1, the first inequality in term 3) assures that spokes do not form links between them while hubs are favorable to form links with each other and spokes are favorable to form links with hubs; the second inequality assures that spokes link with hubs if they acquire partial information actively. Meeting with information level of terms 1) and 2) can guarantee hubs own the aggregate information ŷ₁ in the network, while the aggregate information owned by spokes is ŷ.

We can see from the above situation, after fixing the amount of hubs, the information acquisition of spokes is degressive. So if q ⟶ + ∞, then x_p ⟶ 0. Relatively, the information acquisition of hubs is increasing.

The result also reflects contents of “The law of the few”, that is a lot of information will be grasped in a few hubs, while most of other players, that is, the spokes will choose to link with hubs to get information, but themselves will choose little “personal information acquisition”, or even entirely depends on the connection to get information, making their “personal information acquisition” to zero.

Example 2

Let N = {1, 2, 3, 4, 5, 6, 7, 8, 9} be the set of players, the cost of linking with one other person be homogeneous, we denote by k = 0.15, the cost of information that personally acquired be heterogeneous and is denoted by c₁ = c₂ = c₃ = c₄ = c₅ = c₆ = 12, c₇ = c₈ = c₉ = 1021. Suppose that payoffs are given by (1), where f(y) = ln(1 + y). The initial matrix is a zero matrix, and the initial information vector is (0,0,0,0,0,0,1130,1130,1130).

We can check that s^* = (x^*, g^*) is a Nash equilibrium, where

x∗=(0,0,0,0,0,0,1130,1130,1130),g∗=g1∗g2∗g3∗g4∗g5∗g6∗g7∗g8∗g9∗=000000011000000011000000011000000011000000011000000011000000011000000001000000000

Figure 2

A three-core-completely-periphery equilibrium network

2.2 A Dynamic Formation Procedure and the Algorithm

The algorithm of finding equilibrium networks in this paper is based on a dynamic procedure used in [12, 19, 20].

Given the set of players, at each stage, agents who are chosen at random play their best response, adjust their links in response to the network structure and the personal information acquired by oneself in previous period and then compose the network structure and the personal information (it is called the player’s non-coordinated behavior).

At stage t, first select a subset R of the set of players N at random. For every selected player i ∈ R, we calculate his optimal personal information and the best link set by comparing all of his possible links, namely the player’s best response strategy. Each player who is selected at random chooses a pure strategy best response to the strategy of all other agents in the previous period and for the player who is not selected he maintains the strategy chosen in the previous period. Compose all strategies, and then dynamic procedure comes to stage t + 1.

Based on above dynamic procedure, the advantage of the algorithm is that the optimization question has only one variable. The network reaches a stable state when no one chooses to change his links and personal information during dynamic procedure, and then we can get equilibrium strategy profile s^* = (x^*, g^*) and relative information equilibrium network.

Given an initial stage t₀, an initial information vector xt0=(x1t0,x2t0,⋯,xnt0), initial link matrix [αijt0] ∈ {0, 1}^N×N, where αiit0 = 0, i ∈ N and αijt0 ∈ {0, 1}, ∀ j ∈ N{i}. Homogeneous link costs k ≥ 0 and heterogeneity costs c_i ≥ 0 are given.

At stage t, let information vector be xt=(x1t,x2t,⋯,xnt), link matrix [αijt]=(α1t,α2t,⋯,αnt)T ∈{0, 1}^{N × N}, where αit is the row i of the link matrix [αijt].

Select randomly the set of players R = (i₁, i₂, ⋯, i_r) ⊆ N. For every i_k ∈ R, the row i_k of link matrix has 2ⁿ⁻¹ possibilities if we fix the rest n − 1 rows. For every possibility, first calculate α¯ikjt=max{αikjt,αjikt} ∈ {0, 1}, it is easy to see α¯ikikt=0.

Consider that

subject to xikt ≥ 0.

Notice that ∑j∈Nα¯ikjtxjt and ∑j∈Nαikjtk in objective function are constant for given stage t and all probably value of line i_k in link matrix [αijt]. So the variable in (3) is xikt.

Suppose that the optimal value is reached at xikt. For all the possibilities in row i_k we calculate relative optimal value xikt and θikt. At last by comparing above 2ⁿ⁻¹ optimal value θikt, we can find maxθikt, and then output relative optimal value x~ikt and α~ikt.

Let

be link matrix at stage t + 1, and information vector

The dynamic procedure comes into stage t + 1. Select the set of players at random again and the network formation procedure is repeated.

The dynamic procedure is end until no one chooses to change his links and personal information. At last we get the information flow equilibrium network and equilibrium strategy profile will comprise the link matrix of the equilibrium network and the optimal personal acquired information vector. It should be noticed that to guarantee the valid of the algorithm we must suppose that the dynamic procedure is not a circulation.

Example 3

Given the set of players N = {1, 2, 3, 4, 5, 6}, homogeneous link costs k = 0.04 and personal information heterogeneity costs c₁ = c₂ = c₃ = c₄ = 110, c₅ = c₆ = 111. Payoff function is given by (1), where f(y) = ln(1 + y). The initial link matrix and the initial information vector are 0.

It is different to reach final stage for each implementation of the dynamic process program and convergence to the equilibrium state because the non-coordinated behavior of the players caused by the randomness of process of forming network. Therefore there is not practical significance for number of stages to achieve an equilibrium state.

We only had one interception of several typical stages in some run results in Figure 3, and let K be the number of stages needed to achieve an equilibrium state.

Figure 3

Stage example of the dynamic process

The network structure and dynamics of information vectors in Example 3:

1. At the initial stage we select the set of players R = {2, 3, 4, 5, 6} at random, because of the initial matrix is zero, they all choose to acquire their optimal aggregate information and do not form any link. And then it will come into an empty network by the first round of iteration. But the information level vector is x1=(x11,⋯,x61) = (0, 9, 9, 9, 10, 10).

2. If the selected set of players is R = {3, 5, 6} at stage 1, we can infer that it is the best response for Player 3 to form links with Players 2, 4, 5 and 6 and he do not acquire information personally, the payoff Π₃(.) = ln[1 + (9 + 9 + 10 + 10)] − 4 × 0.04 ≈ 3.5036 is larger than the payoff in any other situation. Because Players 5 and 6 have the same acquired advantage, it is the best response for Players 5 and 6 to form links with 2, 3, 4, 6 and he do not acquire information. The dynamic procedure comes into stage 2.

3. When the dynamic procedure comes into stage t₁, the current network has two-core-periphery architecture. The information level vector of players is x^t₁ = (0, 9, 0, 0, 0, 0). Select set of the players R = {1, 4, 5, 6} randomly. Although here the two-core-periphery architecture is the same as the eventual equilibrium structure formally (Players 2 and 4 are temporal core), obviously, for Players 5 and 6 their aggregate information do not reach their optimal value ŷ₅ = ŷ₆ = 10, so here the network is not an equilibrium network and not stable. For Players 5 and 6 they will delete their links with Player 4 and maintain their link with Player 2 and acquire information 1 by themselves, the payoff Π_i(⋅) = ln[1 + (9 + 1)] − 111 × 1 − 0.04 ≈ 2.2669, i = 5, 6 is lager than in any other situation, so it is the best response for them on this stage. Because Player 4 dose not acquire any information personally, Players 1, 5 and 6 will delete their links with 4 and Players 1 and 4 can acquire their optimal aggregate information 9 by maintaining their links with 2.

4. When the dynamic procedure comes into stage t₂, the current network has three-core-completely-periphery architecture. The information level of players is x^t₂ = (0, 3, 1, 0, 5, 1). Selected set of the players R = N. So here for Players 1 and 4 it is their best response to maintain their links with Players 2, 3, 5 and 6. Players 2 and 3 will maintain their links and reduce their information 1 by themselves and for Players 5 and 6 they maintain their strategies from the previous period. Notice that here the network is four-core-completely-periphery architecture, it is not stable.

5. When the dynamic procedure comes into stage K − 1, the information level of players is x^K−1 = (0, 0, 0, 0, 8, 2). Selected set of the players R = {1} randomly. For Player 1 his best response is to delete his link with Player 3 and maintain his link with 5 and 6 while he acquires no information. The “two-core-periphery” equilibrium network is formed. The hub is composed of two players with information cost advantage acquired actively. The dynamic procedure is over.

Core-periphery structure is the most representative equilibrium network in the case of the local information flow without information loss. In fact, under the premise of players have two levels of information each person who has a comparative advantage of the cost of acquiring information would become the hub possibly. The statistical results show that it is the easiest to form “single-core-periphery” structure; but the “multi-core-periphery” structure which contains all players who has the same advantage on the cost as the hubs (maximum) has the smallest probability; those which between these two kinds become harder and harder with the increasing of the players’ number in the core.

We believe that there are several reasons why the dynamics are important. One of the reasons is that a dynamic model allows us to study the process by which individual agents learn about the network and adjust their links in response to their learning. And the dynamics may help select among different equilibria of the static game.

There is very important prospect to study the characterization of the architecture of information flow equilibrium networks and the dynamics of network formation under the premise of local information flow and without the presence of decay. In addition, consider that some of the players may coordinate and prepare to maximize the common interests of members by some local cooperation; the model and algorithm in this essay can be transformed to an incomplete information cooperative game model. This article assumes that the players have the need and competence of acquiring information personally and getting information from others in the given network structure, in fact, this can also be diversified.